Method of integral equations for dust particles of finite size
52 51
Keywords:
Dusty plasma, screening effects, interaction model, radial distribution function,the static structural factors, the Ornstein-Zernike integral equation, the Percus-Yevick equationAbstract
In this paper we make use of the earlier proposed pseudopotential model of interaction of dusty plasma particles, which correctly takes into account the finite size of the dust particles in the framework of classical plasma electrodynamics in the random-phase approximation. The potential thus constructed differs significantly from the widely used Yukawa (Debye-Hückel) potential at sufficiently large values of the screening parameter, which is explained by the engagement of different boundary conditions at the surface of dust particles. The proposed pseudopotential model is applied to determine the radial distribution functions and the static structural factors of dust particles by the method of integral equations. In particular, the Ornstein-Zernike relation in the hypernetted-chain approximation with the bridge functions for the point-like particles is numerically solved. Since the dimensions of the dust particles are assumed to be finite, the calculations are also carried out within the framework of the combined method of integral equations, which is based on the determination of the correlation functions for the system of hard spheres within the Percus-Yevick equation with further transition to a model of solid charged balls studied within the modified hypernetted-chain approximation. The results show that at high packing fractions, the radial distribution functions and the static structural factors exhibit more significant peaks in comparison with the simple Ornstein-Zernike relation in the hypernetted-chain approximation, which indicates the formation of short-range and long-range orders in the arrangement of dust particles at rather large values of their coupling.
References
1 Fortov V. E., Morfill G. E. Complex and Dusty Plasmas: From Laboratory to Space. – Boca Raton, Florida: CRC Press, 2010. – 440 p.
2 Bonitz M., Horing N., Ludwig P. Introduction to Complex Plasmas. – Salmon Tower Building, New York City: Springer, 2010. – 450 p.
3 Khrapak S., Morfill G. Basic processes in complex (dusty) plasmas: Charging, interaction and ion drag force // Contrib. Plasma Phys. – 2009. – Vol. 49. – P. 148-168.
4 Shukla P. K. and Eliasson B. Colloquium: Fundamentals of dust-plasma interactions // Rev. Mod. Phys. – 2009. – Vol. 81. – P. 25-44.
5 Tsytovich V. N. New paradigm for plasma crystal formation // J. Phys. A: Math. Gen. – 2006. – Vol. 39 – P. 4501-4509.
6 Piel A. Plasma crystals: experiments and simulation // Plasma Phys. Control. Fusion – 2017. – Vol. 59. – 014001 (10 p.).
7 Dietz C., Thoma M. H. Investigation and improvement of three-dimensional plasma crystal analysis // Phys. Rev. E – 2016. – Vol. 94. – 033207 (8 p.).
8 Khlert H. and Bonitz M. How Spherical Plasma Crystals Form // Phys. Rev. Lett. – 2010. – Vol. 104. – P. 015001 (4 p.).
9 Shukla P. K. A survey of dusty plasma physics // Phys. Plasmas – 2001. – Vol. 8. – P. 1791-1803.
10 Fortov V. E., Khrapak A. G., Khrapak S. A., Molotkov V. I., Petrov O. F. Dusty plasmas // Physics-Uspekhi, – 2004. – Vol. 47. – P. 447 -492
11 Seok J. Y., Koo B. C. and Hirashita H. Dust cooling in supernova remnants in the large magellanic cloud // Astrophys. J. – 2015. – Vol. 807. – P. 100-106.
12 Fedoseev A.V., Sukhinin G. I., Abdirakhmanov A. R., Dosbolayev M. K. and Ramazanov T. S. Voids in Dusty Plasma of a Stratified DC Glow Discharge in Noble Gases // Contrib. Plasma Phys. – 2016. – Vol. 56. – P. 234-239.
13 Tolias P., Ratynskaia S., de Angeli M., de Temmerman G., Ripamonti D., Riva G., Bykov I., Shalpegin A., Vignitchouk L., Brochard F., Bystrov K., Bardin S., Litnovsky A. Dust remobilization in fusion plasmas under steady state conditions // Plasma Phys. Control. Fusion – 2016. – Vol. 58. – P. 025009 (16 p.).
14 Castaldo C., Ratynskaia S., Pericolli V., de Angelis U., Rypdal K., Pieroni L., Giovannozzi E., Mad-daluno G., Marmolino C., Rufoloni A., Tuccillo A., Kretschmer M. and Morfill G. E. Diagnostics of fast dust particles in tokamak edge plasmas // Nucl. Fusion – 2007. – Vol. 47. – P. L5-L9.
15 Keidar M., Shashurin A., Volotskova O., Stepp M. A., Srinivasan P., Sandler A. and Trink B. Cold atmospheric plasma in cancer therapy // Phys. Plasmas – 2013. – Vol. 20. – P. 057101 (8 p.).
16 Walk R. M., Snyder J. A., Scrivasan P., Kirch J., Diaz S. O., Blanco F. C., Shashurin A., Keidar M. and Sandler A. D. Cold atmospheric plasma for the ablative treatment of neuroblastoma // J. Pediatr. Surg. – 2013. – Vol. 48. – P. 67-73.
17 Yousefi R., Davis A.B., Carmona-Reyes J., Matthews L.S., Hyde T.W. Measurement of net electric charge and dipole moment of dust aggregates in a complex plasma // Phys. Rev. E. – 2014. – Vol. 90. – P. 033101 (6 p.).
18 Ramazanov T.S., Bastykova N.Kh., Ussenov Y.A., Kodanova S.K., Dzhumagulova K.N., Dosbolayev M.K. The Behavior of Dust particles Near Langmuir Probe // Contrib. Plasma Phys. – 2012. – Vol. 52. – P. 110-113.
19 Bonitz M., Henning C., and Block D. Complex plasmas: a laboratory for strong correlations // Rep. Prog. Phys. – 2010. – Vol. 73. – P. 066501 (29 p.).
20 Kalman G., Hartmann P., Donko Z., Golden K. J., Kyrkos S. Collective modes in two-dimensional binary Yukawa systems // Phys. Rev. E – 2013. – Vol. 87. – P. 043103 (15 p.).
21 Khrapak S. A., Thomas H. M., Fluid approach to evaluate sound velocity in Yukawa systems and complex plasmas // Phys. Rev. E – 2015. – Vol. 91. – P. 033110 (8 p.).
22 Khrapak S. A., Thomas H. M., Filinov, V. S. and Fortov, V. E. and Bonitz, M. and Moldabekov, Zh. Fermionic path-integral Monte Carlo results for the uniform electron gas at finite temperature // Phys. Rev. E – 2015. – Vol. 91. – P. 033108 (12 p.).
23 Davletov A.E., Arkhipov Yu.V., Tkachenko I.M. Electric charge of dust particles in a plasma// Contrib. Plasma Phys. – 2016. – Vol. 56. – P. 308.
24 Iyetomi H., Ogata S., Ichimaru S. Bridge functions and improvement on the hypernetted-chain approximation for classical one-component plasmas// Phys. Rev. A – 1992. – Vol. 46. – P. 1051.
25 Daughton W., Murillo M.S., Thode L. Empirical bridge function for strongly coupled Yukawa systems// Phys. Rev. E – 2000. – Vol. 61. – P. 2129.
26 Wertheim M.S. Exact solution of the Percus-Yevick integral equation for hard spheres // Phys. Rev. Lett. – 1963. – Vol. 10, No 8. – P. 321.
27 Lado F. //Mol. Phys.– 1976. – Vol. 31. – P. 1117.
28 Lado F., Foiles S., Ashcroft N.W. Solutions of the reference-hypernetter-chain equation with minimized free energy// Phys. Rev. A – 1983. – Vol. 28. – P. 2374
29 Rosenfeld Y. Comment of the variational modified-hypernetted-chain theory for simple liquids// J. Stat. Phys. – 1986. – Vol. 42. – P. 437
References
1 V.E. Fortov and G.E. Morfill, Complex and Dusty Plasmas: From Laboratory to Space (CRC Press, Florida, USA, 2010), 440 p.
2 M. Bonitz and N. Horing, P. Ludwig, Introduction to Complex Plasmas (Springer Publishing, New York, 2010), 450 p.
3 S. Khrapak and G. Morfill, Contrib. Plasma Phys. 49, 148-168, (2009).
4 P.K. Shukla and B. Eliasson, Rev. Mod. Phys. 81, 25-44, (2009).
5 V.N. Tsytovich, J. Phys. A: Math. Gen. 39, 4501-4509, (2006).
6 A. Piel, Plasma Phys. Control. Fusion 59, 014001, (2017).
7 C. Dietz and M.H. Thoma, Phys. Rev. E. 94, 033207, (2016).
8 H. Khlert and M. Bonitz, Phys. Rev. Lett. 104, 015001, (2010).
9 P.K. Shukla, Phys. Plasmas 8, 1791-1803, (2001).
10 V.E. Fortov, A.G. Khrapak, S.A. Khrapak, V.I. Molotkov, and O.F. Petrov, Physics-Uspekhi 47, 447-492 (2004).
11 J.Y. Seok, B.C. Koo and H. Hirashita, Astrophys. J. 807, 100-106, (2015).
12 A.V. Fedoseev, G.I. Sukhinin, A.R. Abdirakhmanov, M.K. Dosbolayev and T.S. Ramazanov, Contrib. Plasma Phys. 56, 234-239, (2016).
13 P. Tolias, S. Ratynskaia, M. de Angeli, G. de Temmerman, D. Ripamonti, G. Riva, I. Bykov, A. Shalpegin, L.Vignitchouk, F. Brochard, K. Bystrov, S. Bardin, and A. Litnovsky, Plasma Phys. Control. Fusion 58, 025009, (2016).
14 C. Castaldo, S. Ratynskaia, V. Pericolli, U. de Angelis, K. Rypdal, L. Pieroni, E. Giovannozzi, G. Mad-dalu no, C. Marmolino, A. Rufoloni, A. Tuccillo, M. Kretschmer and G.E. Morfill, Nucl. Fusion 47, L5-L9, (2007).
15 M. Keidar, A. Shashurin, O. Volotskova, M.A. Stepp, P. Srinivasan, A. Sandler and B. Trink, Phys. Plasmas 20, 057101, (2013).
16 R.M. Walk, J.A. Snyder, P. Scrivasan, J. Kirch, S.O. Diaz, F.C. Blanco, A. Shashurin, M. Keidar and A.D. Sandler, J. Pediatr. Surg. 48, 67-73, (2013).
17 R. Yousefi, A.B. Davis, J. Carmona-Reyes, L.S. Matthews, and T.W. Hyde, Phys. Rev. E. 90, 033101, (2014).
18 T.S. Ramazanov, N.Kh. Bastykova, Y.A. Ussenov, S.K. Kodanova, K.N. Dzhumagulova, and M.K. Dosbolayev, Contrib. Plasma Phys. 52, 110-113, (2012).
19 M. Bonitz, C. Henning, and D. Block, Rep. Prog. Phys. 73, 066501, (2010).
20 G. Kalman, P. Hartmann, Z. Donko, K.J. Golden, and S. Kyrkos, Phys. Rev. E. 87, 043103, (2013).
21 S.A. Khrapak and H.M. Thomas, Phys. Rev. E. 91, 033110, (2015).
22 S.A. Khrapak, H.M. Thomas, V.S. Filinov, V.E. Fortov, M. Bonitz, and Zh. Moldabekov, Phys. Rev. E 91, 033108, (2015).
23 A.E. Davletov, Yu.V. Arkhipov, and I.M. Tkachenko, Contrib. Plasma Phys. 56, 308 (2016).
24 H. Iyetomi, S. Ogata, and S. Ichimaru, Phys. Rev. A 46, 1051, (1992).
25 W. Daughton, M.S. Murillo, and L. Thode, Phys. Rev. E. 61, 2129, (2000).
26 M.S. Wertheim, Phys. Rev. Lett. 10, 8, 321, (1963).
27 F. Lado Mol. Phys. 31, 1117, (1976).
28 F. Lado, S. Foiles, and N.W. Ashcroft, Phys. Rev. A. 28, 2374, (1983).
29 Y. Rosenfeld, J. Stat. Phys. 42, 437, (1986).
2 Bonitz M., Horing N., Ludwig P. Introduction to Complex Plasmas. – Salmon Tower Building, New York City: Springer, 2010. – 450 p.
3 Khrapak S., Morfill G. Basic processes in complex (dusty) plasmas: Charging, interaction and ion drag force // Contrib. Plasma Phys. – 2009. – Vol. 49. – P. 148-168.
4 Shukla P. K. and Eliasson B. Colloquium: Fundamentals of dust-plasma interactions // Rev. Mod. Phys. – 2009. – Vol. 81. – P. 25-44.
5 Tsytovich V. N. New paradigm for plasma crystal formation // J. Phys. A: Math. Gen. – 2006. – Vol. 39 – P. 4501-4509.
6 Piel A. Plasma crystals: experiments and simulation // Plasma Phys. Control. Fusion – 2017. – Vol. 59. – 014001 (10 p.).
7 Dietz C., Thoma M. H. Investigation and improvement of three-dimensional plasma crystal analysis // Phys. Rev. E – 2016. – Vol. 94. – 033207 (8 p.).
8 Khlert H. and Bonitz M. How Spherical Plasma Crystals Form // Phys. Rev. Lett. – 2010. – Vol. 104. – P. 015001 (4 p.).
9 Shukla P. K. A survey of dusty plasma physics // Phys. Plasmas – 2001. – Vol. 8. – P. 1791-1803.
10 Fortov V. E., Khrapak A. G., Khrapak S. A., Molotkov V. I., Petrov O. F. Dusty plasmas // Physics-Uspekhi, – 2004. – Vol. 47. – P. 447 -492
11 Seok J. Y., Koo B. C. and Hirashita H. Dust cooling in supernova remnants in the large magellanic cloud // Astrophys. J. – 2015. – Vol. 807. – P. 100-106.
12 Fedoseev A.V., Sukhinin G. I., Abdirakhmanov A. R., Dosbolayev M. K. and Ramazanov T. S. Voids in Dusty Plasma of a Stratified DC Glow Discharge in Noble Gases // Contrib. Plasma Phys. – 2016. – Vol. 56. – P. 234-239.
13 Tolias P., Ratynskaia S., de Angeli M., de Temmerman G., Ripamonti D., Riva G., Bykov I., Shalpegin A., Vignitchouk L., Brochard F., Bystrov K., Bardin S., Litnovsky A. Dust remobilization in fusion plasmas under steady state conditions // Plasma Phys. Control. Fusion – 2016. – Vol. 58. – P. 025009 (16 p.).
14 Castaldo C., Ratynskaia S., Pericolli V., de Angelis U., Rypdal K., Pieroni L., Giovannozzi E., Mad-daluno G., Marmolino C., Rufoloni A., Tuccillo A., Kretschmer M. and Morfill G. E. Diagnostics of fast dust particles in tokamak edge plasmas // Nucl. Fusion – 2007. – Vol. 47. – P. L5-L9.
15 Keidar M., Shashurin A., Volotskova O., Stepp M. A., Srinivasan P., Sandler A. and Trink B. Cold atmospheric plasma in cancer therapy // Phys. Plasmas – 2013. – Vol. 20. – P. 057101 (8 p.).
16 Walk R. M., Snyder J. A., Scrivasan P., Kirch J., Diaz S. O., Blanco F. C., Shashurin A., Keidar M. and Sandler A. D. Cold atmospheric plasma for the ablative treatment of neuroblastoma // J. Pediatr. Surg. – 2013. – Vol. 48. – P. 67-73.
17 Yousefi R., Davis A.B., Carmona-Reyes J., Matthews L.S., Hyde T.W. Measurement of net electric charge and dipole moment of dust aggregates in a complex plasma // Phys. Rev. E. – 2014. – Vol. 90. – P. 033101 (6 p.).
18 Ramazanov T.S., Bastykova N.Kh., Ussenov Y.A., Kodanova S.K., Dzhumagulova K.N., Dosbolayev M.K. The Behavior of Dust particles Near Langmuir Probe // Contrib. Plasma Phys. – 2012. – Vol. 52. – P. 110-113.
19 Bonitz M., Henning C., and Block D. Complex plasmas: a laboratory for strong correlations // Rep. Prog. Phys. – 2010. – Vol. 73. – P. 066501 (29 p.).
20 Kalman G., Hartmann P., Donko Z., Golden K. J., Kyrkos S. Collective modes in two-dimensional binary Yukawa systems // Phys. Rev. E – 2013. – Vol. 87. – P. 043103 (15 p.).
21 Khrapak S. A., Thomas H. M., Fluid approach to evaluate sound velocity in Yukawa systems and complex plasmas // Phys. Rev. E – 2015. – Vol. 91. – P. 033110 (8 p.).
22 Khrapak S. A., Thomas H. M., Filinov, V. S. and Fortov, V. E. and Bonitz, M. and Moldabekov, Zh. Fermionic path-integral Monte Carlo results for the uniform electron gas at finite temperature // Phys. Rev. E – 2015. – Vol. 91. – P. 033108 (12 p.).
23 Davletov A.E., Arkhipov Yu.V., Tkachenko I.M. Electric charge of dust particles in a plasma// Contrib. Plasma Phys. – 2016. – Vol. 56. – P. 308.
24 Iyetomi H., Ogata S., Ichimaru S. Bridge functions and improvement on the hypernetted-chain approximation for classical one-component plasmas// Phys. Rev. A – 1992. – Vol. 46. – P. 1051.
25 Daughton W., Murillo M.S., Thode L. Empirical bridge function for strongly coupled Yukawa systems// Phys. Rev. E – 2000. – Vol. 61. – P. 2129.
26 Wertheim M.S. Exact solution of the Percus-Yevick integral equation for hard spheres // Phys. Rev. Lett. – 1963. – Vol. 10, No 8. – P. 321.
27 Lado F. //Mol. Phys.– 1976. – Vol. 31. – P. 1117.
28 Lado F., Foiles S., Ashcroft N.W. Solutions of the reference-hypernetter-chain equation with minimized free energy// Phys. Rev. A – 1983. – Vol. 28. – P. 2374
29 Rosenfeld Y. Comment of the variational modified-hypernetted-chain theory for simple liquids// J. Stat. Phys. – 1986. – Vol. 42. – P. 437
References
1 V.E. Fortov and G.E. Morfill, Complex and Dusty Plasmas: From Laboratory to Space (CRC Press, Florida, USA, 2010), 440 p.
2 M. Bonitz and N. Horing, P. Ludwig, Introduction to Complex Plasmas (Springer Publishing, New York, 2010), 450 p.
3 S. Khrapak and G. Morfill, Contrib. Plasma Phys. 49, 148-168, (2009).
4 P.K. Shukla and B. Eliasson, Rev. Mod. Phys. 81, 25-44, (2009).
5 V.N. Tsytovich, J. Phys. A: Math. Gen. 39, 4501-4509, (2006).
6 A. Piel, Plasma Phys. Control. Fusion 59, 014001, (2017).
7 C. Dietz and M.H. Thoma, Phys. Rev. E. 94, 033207, (2016).
8 H. Khlert and M. Bonitz, Phys. Rev. Lett. 104, 015001, (2010).
9 P.K. Shukla, Phys. Plasmas 8, 1791-1803, (2001).
10 V.E. Fortov, A.G. Khrapak, S.A. Khrapak, V.I. Molotkov, and O.F. Petrov, Physics-Uspekhi 47, 447-492 (2004).
11 J.Y. Seok, B.C. Koo and H. Hirashita, Astrophys. J. 807, 100-106, (2015).
12 A.V. Fedoseev, G.I. Sukhinin, A.R. Abdirakhmanov, M.K. Dosbolayev and T.S. Ramazanov, Contrib. Plasma Phys. 56, 234-239, (2016).
13 P. Tolias, S. Ratynskaia, M. de Angeli, G. de Temmerman, D. Ripamonti, G. Riva, I. Bykov, A. Shalpegin, L.Vignitchouk, F. Brochard, K. Bystrov, S. Bardin, and A. Litnovsky, Plasma Phys. Control. Fusion 58, 025009, (2016).
14 C. Castaldo, S. Ratynskaia, V. Pericolli, U. de Angelis, K. Rypdal, L. Pieroni, E. Giovannozzi, G. Mad-dalu no, C. Marmolino, A. Rufoloni, A. Tuccillo, M. Kretschmer and G.E. Morfill, Nucl. Fusion 47, L5-L9, (2007).
15 M. Keidar, A. Shashurin, O. Volotskova, M.A. Stepp, P. Srinivasan, A. Sandler and B. Trink, Phys. Plasmas 20, 057101, (2013).
16 R.M. Walk, J.A. Snyder, P. Scrivasan, J. Kirch, S.O. Diaz, F.C. Blanco, A. Shashurin, M. Keidar and A.D. Sandler, J. Pediatr. Surg. 48, 67-73, (2013).
17 R. Yousefi, A.B. Davis, J. Carmona-Reyes, L.S. Matthews, and T.W. Hyde, Phys. Rev. E. 90, 033101, (2014).
18 T.S. Ramazanov, N.Kh. Bastykova, Y.A. Ussenov, S.K. Kodanova, K.N. Dzhumagulova, and M.K. Dosbolayev, Contrib. Plasma Phys. 52, 110-113, (2012).
19 M. Bonitz, C. Henning, and D. Block, Rep. Prog. Phys. 73, 066501, (2010).
20 G. Kalman, P. Hartmann, Z. Donko, K.J. Golden, and S. Kyrkos, Phys. Rev. E. 87, 043103, (2013).
21 S.A. Khrapak and H.M. Thomas, Phys. Rev. E. 91, 033110, (2015).
22 S.A. Khrapak, H.M. Thomas, V.S. Filinov, V.E. Fortov, M. Bonitz, and Zh. Moldabekov, Phys. Rev. E 91, 033108, (2015).
23 A.E. Davletov, Yu.V. Arkhipov, and I.M. Tkachenko, Contrib. Plasma Phys. 56, 308 (2016).
24 H. Iyetomi, S. Ogata, and S. Ichimaru, Phys. Rev. A 46, 1051, (1992).
25 W. Daughton, M.S. Murillo, and L. Thode, Phys. Rev. E. 61, 2129, (2000).
26 M.S. Wertheim, Phys. Rev. Lett. 10, 8, 321, (1963).
27 F. Lado Mol. Phys. 31, 1117, (1976).
28 F. Lado, S. Foiles, and N.W. Ashcroft, Phys. Rev. A. 28, 2374, (1983).
29 Y. Rosenfeld, J. Stat. Phys. 42, 437, (1986).
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Yerimbetova, L. (2017). Method of integral equations for dust particles of finite size. Recent Contributions to Physics (Rec.Contr.Phys.), 62(3), 40–49. Retrieved from https://bph.kaznu.kz/index.php/zhuzhu/article/view/557
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